Vortrag im Seminar für Kombinatorik und Optimierung

Jigsaw percolation, introduced by Brummit, Chatterjee, Dey and Sivakoff, may be viewed as a measure of whether two graphs on the same vertex set are ``jointly connected'': Given two such graphs, we ``merge'' two vertices if they are connected in both graphs, and the new vertex inherits all of the neighbours of the old vertices in both graphs. We continue until no more vertices can be merged and say the process percolates} if it ends with a single vertex.

Bollob\'as and Riordan proved that if the two graphs are independent binomial random graphs $G(n,p_1)$ and $G(n,p_2)$, then there is a threshold for percolation based on the product $p_1p_2$ of the two edge probabilities. However, they only determined the location of this threshold up to a constant factor.

In this talk we will present work in progress towards finding the exact threshold. This is based on joint work with Tobias Kapetanopoulos, Tam\'as Makai and Kathrin Skubch.

Vortrag im Seminar für Kombinatorik und Optimierung

This talk will be a gentle introduction to the field of enumerative combinatorics.

Generating functions are a standard tool in enumerative combinatorics. In order to count the objects in a given class of combinatorial structures, the numbers of objects of different sizes are represented in a formal power series. Constructive decompositions of the objects in the class correspond to certain functional operations for the power series. Tools for analysing such operations allow to determine the exact or asymptotic numbers of objects in the class. However, if one wants to count objects up to some type of isomorphism, this approach usually fails.

In this talk, we present a different formal representation of combinatorial classes that takes isomorphisms into account: cycle index sums. These formal sums represent not only the objects in a given class, but also the orbits of their automorphisms, which makes it possible to represent constructive decompositions by functional operations that depend on the orbit.

Strukturtheorie-Seminar

I will describe the limit moments of a random matrix of Wishart type $BB^*$, where $B$ is a product of (classically) independent rectangular random matrices, using combinatorics related to matricially free probability. Those moments will be expressed in terms of noncrossing partitions with even blocks related to certain words, and then using $R$-, $S$- and $T$-transforms well known in free probability we give an explicit formula for those moments, connecting them with the multivariate Fuss-Narayana polynomials. At the end I'll present some partial results without the assumption of independence of random matrices. The results are joint work with R. Lenczewski.

Vortrag im Seminar für Kombinatorik und Optimierung

Bootstrap percolation on a graph, with infection threshold a positive integer $r$, is an infection process which starts out from a set of initially infected vertices and in each further step every vertex with at least $r$ infected neighbours becomes infected. The process stops once no further vertices can become infected.

We consider bootstrap percolation on the binomial random graph $G(n,p)$, when the set of initially infected vertices has size $a$, which was investigated by among others by Janson, \L uczak, Turova and Valier (2012). We improve their results by strengthening the probability bounds for the number of infected vertices at the end of the process.

Abstract: Let A be an abelian variety over a number field K. We are
interested in its group of K-rational points A(K), which is finitely
generated by the Mordell-Weil Theorem. We show how to provide an explicit
upper bound on the rank of A(K) in terms of the height of A and the
discriminant of K. The proof uses a new inequality between the height and
bad reduction primes, which is obtained by combining a jacobian argument,
a Bertini Theorem and the existence of unramified towers over certain
quadratic number fields.

Abstract: We explore in depth the number theoretic and statistical properties of certain sets of numbers arising from their Cantor series expansions. As a direct consequence of our main theorem we deduce numerous new results as well as strengthen known ones.

Vortrag im Seminar für Kombinatorik und Optimierung

For integers $g,m\ge 0$ and $n>0$, let $S^g_{n,m}$ denote the graph taken uniformly at random from the set of all graphs on $\{1,2, \ldots , n\}$ with exactly $m=m(n)$ edges and with genus at most $g$. We use counting arguments to investigate the components, subgraphs, maximum degree, and largest face size of $S^g_{n,m}$, finding that there is often different asymptotic behaviour depending on the ratio $m/n$.

We propose exact solution approaches for two different
generalization of the 0-1 Knapsack Problem, that is the
0-1 Knapsack Problem with Setups and the 0-1 Collapsing
Knapsack Problem. In the first problem, the items belong
to disjoint families (or classes) and they can be picked
only if the corresponding family is activated. The
selection of a class involves setup costs and resource
consumptions thus affecting both the objective function
and the capacity constraint. In the 0-1 Collapsing
Knapsack Problem, the capacity of the knapsack is not a
scalar but a non-increasing function of the number of
items included, namely it is inversely related to the
number of items placed inside the knapsack. The solution
approaches we propose rely on an effective exploration of
a specific set of variables that leads to solve standard
knapsack problems and on dynamic programming. For the
Knapsack Problem with Setups, a series of approximation
results is also provided.

We develop the asymptotic formulas for correlations
\[
\sum_{n\le x}f_1(P_1(n))f_2(P_2(n))\cdot \dots \cdot f_m(P_m(n))\] where $f_1,\dots,f_m$ are bounded ``pretentious'' multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences: first, we characterize all multiplicative functions $f:\mathbb{N}\to\{-1,+1\}$ with bounded partial sums. This answers a question of Erd\H{o}s from $1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either $f(n)=n^s$ for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of K\'atai. If time permits, we discuss multidimensional analog of the results above and its application to the study of Gowers norms of multiplicative functions. Finally, we apply these techniques to study sign patterns of $(f(n),f(n+1),f(n+2))$ and $(f(n),f(n+1),f(n+2),f(n+3))$ where $f:\mathbb{N}\to \{-1,1\}$ is a given multiplicative function.

FoE-Kolloquium

Phase retrieval refers to restoring a signal from intensity measurements
only and is important in several application areas ranging from X-ray crystallography and microscopy to audio processing and deep learning. Due to the substantial amount of missing information, phase retrieval problems are extremely challenging to solve numerically.

We shed some light on the origin of these difficulties and we propose a novel paradigm of weaker stability which actually achieves stable phase retrieval in several applications.

The Turan function ex$(n,F)$ of a graph $F$ is the maximum number of edges in an $F$-free graph with $n$ vertices. The results of Turan and Rademacher from 1941 led to the study of supersaturated graphs where the key question is to determine the minimum number of copies of $F$ that a graph with n vertices and ex$(n,F)+q$ edges can have. In general, this problem seems to be very difficult and is wide open.

I will present joint results with Zed Yilma, where we concentrate on the case when $q=o(n^2)$ and $F$ is colour-critical (that is, the removal of some edge from $F$ decreases its chromatic number).

The classic analysis of the probability that the Galton-Watson tree is infinite gives an equation with two solutions. What about other properties. For first order properties we show that the equation system has a unique solution. When conditioning on the tree being infinite the probabilities of first order properties are particularly nice. More generally we consider properties defined by tree automata. These are associated with Monadic Second Order Logic. There is then a natural equation system. Sometimes the system has rogue solutions, meaning solutions with no interpretation. This area combines structural combinatorics, probabilistic combinatorics and logic. (Joint work with Moumanti Podder.)

FWF START Seminar

The hexagonal lattice gives the highest density circle packing among all lattices in the plane.
In this talk I first recall the basic notions about lattices in the plane, before I construct a sequence of lattices with integer bases that approximate the hexagonal lattice. The construction uses elementary number theory and is based on particular palindromic continued fraction expansions. As an application I obtain lattices modulo N with longest possible shortest distances.

14:00 {\bf H. Edelsbrunner} (ISTA): {\bf Distances and divergences in
topological data analysis}.
Given a finite set in a metric space, the topological
analysis assesses its multi-scale connectivity quantified in terms
of a $1$-parameter family of homology groups. Going beyond metrics,
we show that the basic tools of topological data analysis also
apply when we measure dissimilarity with Bregman divergences.
A particularly interesting case is the relative entropy whose
infinitesimal version is known as the Fisher information metric.
It relates to the Euclidean metric on the sphere and, perhaps
surprisingly, the discrete Morse properties of random data
behaves the same as in Euclidean space.

{\bf Coffee Break}

15:30 {\bf Raman Sanyal} (Goethe-Univ.\ Frankfurt): {\bf The geometry of double posets}, In 1984 Stanley showed that partially ordered sets (posets)
can be studied from the perspective of geometry. His order- and chain
polytopes express combinatorial properties of posets as
geometric quantities, furnishing a number of deep results
in both combinatorics and geometry.
In 2011, Malvenuto and Reutenauer introduced double posets (finite
sets equipped with two partial order relations). This notion
underlies many constructions in algebraic combinatorics such as P-partitions,
Littlewood-Richardson rules, permutation statistics.
In this talk, I will explain how double posets can be studied from a geometric
point of view, highlighting connections to
centrally-symmetric polytopes (weak Hanner polytopes), combinatorial
optimization (anti-blocking polytopes), valuations on distributive lattices,
and generalized Hibi (semigroup) rings. This is joint work with T.\ Chappell
and T.\ Fried.

16:20 {\bf Wöden Kusner} (TU Graz): {\bf Critical packings, rigidity, and the radius function}.
There are a number of classical problems in geometric optimization that ask
for the ``best'' configuration of points with respect to some function. We are
interested in the relationships between various notions of criticality for
such functions on configuration spaces, in particular the injectivity or
packing radius. This is not a Morse function, but it has been observed to
be Morse-like, in that the topological notion of regularity can be defined
in an analogous way. Furthermore, there is a geometric interpretation from
rigidity theory that characterizes configurations as critical by the
existence of a strut measure.

Gastvortrag

The Nerve Theorem is an implicit tool in most application of topological data analysis relating the topological type of a suitably nice space with a combinatorial description of the space, namely, the nerve of a cover of that space. It is required that it is a good cover, that each element and intersection is contractible or at least acyclic. In this talk, I will describe a weaker condition we call an epsilon-acyclic cover. It encodes the idea that if a cover is almost a good cover, the persistent homology of a filtration computed on the nerve is a good approximation of the persistent homology of a filtration on the underlying space. The main application of this result is to reduce the computational burden for computing persistence by allowing the use of coarser representations of the space (e.g. smaller simplicial complexes). I will also describe how to obtain explicit error bounds from local computations.

Gastvortrag

I will discuss the efficient computation of the persistence barcode of a Vietoris--Rips filtration given a finite metric space. The implementation in the developed C++ code “Ripser” focuses on memory and time efficiency, where the code outperforms existing software by a factor of more than 40 in computation time and a factor of more than 15 in memory efficiency.

Motivated by real-time logistics, I will present a deterministic algorithm for the Online Minimum Metric Bipartite Matching Problem. In this problem, we are given a set S of server locations and a set R of request locations. The requests arrive one at a time and when it arrives, we must immediately and irrevocably match it to a ``free" server. The cost of matching a server to request is given by the distance between the two locations (which we assume satisfies triangle inequality). The objective of this problem is to come up with a matching of servers to requests which is competitive with respect to the minimum-cost matching of S and R. The algorithm that I present is based on a variant of the offline primal-dual method where the constraint for every edge is relaxed by a fixed multiplicative factor. Using the fact that the costs between locations satisfy triangle inequality, I will show that this deterministic algorithm simultaneously achieves optimal performances in two well-known online models -- the adversarial and the random arrival models.

Bio: Dr. Sharath Raghvendra is an Assistant Professor in the Department of Computer Science at Virginia Tech. Before joining Virginia Tech, he spent two years as a postdoctoral scholar at Stanford University. Dr. Raghvendra received his Ph.D from Duke University in 2012 where he won the Best Doctoral Dissertation Award. He is also the recipient of NSF CRII Award in 2014. His research interests span design of algorithms for problems in computational geometry and graph theory.

Let $d\geq 2$ be an integer which is not a square. Let $(X_n,Y_n)_{n\ge 1}$ be the sequence of positive integer solutions $(X,Y)$ of the Pell equation $X^2-dY^2=c$, where $c\in \{\pm 1,\pm 4\}$. We then show that there is at most one
$n$ such that $X_n$ is a rep-digit in base $10$, with a few exceptions in $d$. The same result holds if we impose that $X_n$ is a rep-digit in any fixed integer base $b\ge 2$ or a member of the Fibonacci sequence. The proofs use linear forms in logarithms of algebraic numbers.

Gastvortrag

For over two decades, Exact Geometric Computation (EGC) has provided a paradigm in Computational Geometry for the correct implementation of geometric algorithms. It is the most successful approach for addressing numerical nonrobustness, leading to successful libraries and practical algorithms
in many areas. We review some pressing reasons to extend this paradigm:

Some common themes emerge from the above list: we replace the exact algebraic model of computation (Real RAM or BSS Model) by a model based on numerical interval approximations. The main algorithmic paradigms are subdivision and iteration. We introduce an input resolution parameter (epsilon), but it must
be used in a novel ``soft way''. Soft versions of classical (``hard'')
geometric predicates must be developed.

What are the consequences of such a computational paradigm ?

* scope of computational geometry would be vastly broadened

* some unsolvable problems in the Real RAM model becomes possible

* soft algorithms are not only implementable but can be practical

One challenge is to revisit classical problems of computational
geometry from this new view point. Another challenge is to
produce complexity analysis of such algorithms.

Strukturtheorie-Seminar

For an iterated function system, the points of the attractor can be encoded by infinite words on a finite alphabet. Typically, the associated Cayley graph forms a tree; but when it does not, topological properties of the attractor can be deduced from the graph itself by using methods from geometric group theory. In particular, if the graph has only one end, the attractor is connected.

Groups of infinite words also appear in the algebraic description of wild topological spaces, such as the Hawaiian earring and the harmonic archipelago. Using a variety of tools (combinatorial dynamics, infinite systems of equations, classical abelian group theory), we highlight some surprising aspects of their homology and their fundamental groups.

We will discuss the Ore condition for group rings in the context of the open problem of embedding group algebras of torsion free groups into skew-fields. We will report on the recent result joint with L. Bartholdi which confirms the conjectured equivalence of Ore condition for group algebras with amenability of the underlying group.

We will then proceed to highlight a connection between the Ore condition and
the free calculus of Fox, and see how that in turn relates to 3-manifold invariants and $L^2$ homology via deteminants of matrices over a group algebra. These last results were obtained jointly with F. Funke.

Structure theory and stochastics seminar

IDLA is a random growth model on a graph, in which particles move randomly around until finding an empty site where they stop and stick to an existing initial cluster. One of the main questions here is to understand the shape of the (random) growing cluster of occupied sites, when we start a large number of particles at the origin (or any fixed vertex) of our graph. I will first describe briefly the model, the connection with other models, and the state of the art. Then I will state some new results about the shape of IDLA on Sierpinski gasket graphs.
This is based on a joint work (in progress) with Joe Chen, Wilfried Huss and Alexander Teplyaev.

Structure theory and stochastics seminar

The construction of a Brownian motion on a self-similar fractal was intensely studied in the 80s and 90s. For many fractals, such as the Sierpinski gasket, the approach is to consider (single-particle) random walks on the graphs approximating the fractal, rescale space and time, and then prove that a scaling limit exist.

Now consider, on a fractal graph, multiple random walkers which interact with each other according to some local rules. Fix an initial density of random walkers at time 0, and evolve the system in time. Can one prove a scaling limit of the time-evolving particle densities along successive graph approximations of the fractal? And if so, what is the limit object?

We answer this question positively in the setting of the (weakly asymmetric) exclusion process on the Sierpinski gasket. In this case, the scaling is the same as the diffusive scaling for single-particle random walks, and the limit of the particle densities evolves according to a (nonlinear) heat equation (a law of large numbers result). We can also characterize the fluctuations about the mean of the density (a large deviations result). In this talk, I will describe what these results mean precisely, and mention one or two key technical tools used to prove them.

The program starts on Thursday, 30. June at 10:00 (coffee break), followed by the talk by M. Beiglböck (10:30), and continues with contributed talks.

On Friday, 1. July, the program starts at 9:00 with contributed talks; the invited talk by R. Grübel is at 10:30 and is also part of the ``Advanced Topics'' seminar of the DK ``Discrete Mathematics''.

For details, see

www.math.tugraz.at/mathc/stochdays/

Vortrag im Seminar für Kombinatorik und Optimierung

Title: Law of large numbers for the largest component of random graphs on the hyperbolic planeSpeaker: Nikolaos Fountoulakis (University of Birmingham)Date: Dienstag 28.6.2016, 14:15Room: Seminarraum AE06, Steyrergasse 30, Erdgeschoss

Abstract:

In this talk, we consider a recent model which was developed by Krioukov et al. of random geometric graphs on the hyperbolic plane. This may be also viewed as the geometrisation of the well-known Chung-Lu model of inhomogeneous random graphs. We show that the size of the largest connected component undergoes a phase transition and a giant component emerges as soon as the curvature of the underlying hyperbolic plane crosses a certain value. We also show that the fraction of vertices that are contained there converges in probability to a certain constant, which is related to a continuum percolation model on the upper-half plane.

The talk will be about using geometric data structures to solve the
bottleneck and linear assignment problems in geometric graphs. The motivation is
computational topology, where these problems appear for persistence diagrams, which are subsets of the plane, so we are going to deal with bipartite graphs embedded in $\mathbb{R}^2$ with edge cost being some power of the distance between the vertices. The main underlying structure in both problems will be k-d tree (with some modifications for the linear assignment problem).

Strukturtheorie-Seminar

A brief overview will be given of how graphs can be used in the theory of topological groups and how results about topological groups can be applied in the study of automorphism groups of graphs. In particular, the connections between the minimal valency of a so called Abels-Cayley graph for a totally disconnected locally compact group and the structure of the groups is explained.

Vortrag im Seminar Kombinatorik und Optimierung

The classical bin packing problem requires to pack n elements with sizes $a_i$, $0\le a_i\le 1$ for $i=1,....,n$, into a minimum number of unit capacity bins. The problem is well known to be NP-hard. Therefore a lot of (polynomial time) approximation algorithms have been defined and analysed. We will consider the class of online algorithms where the items appear one after the other, and the algorithm has to place the items immediately and irrevocably without knowing anything about the subsequent items. One possibility to measure the effciency of an algorithm to give its asymptotic competitive ratio (ACR) which is given
as follows. Let $A$ be an arbitrary approximation algorithm, and let $I$ be an instance of the given problem. Let A($I$) and OPT($I$) denote the solution of $A$ and the optimal solution, respectively. The ACR is defned as follows:
$\displaystyle R_A^\infty = \limsup_{k\to \infty} \,\{ \max_{I} \{ A(I)/k: \mbox{\rm OPT}(I)=k\}\}$.

It is clear that online algorithms do not have enough information to produce a good packing. So the question in hand: how good can an online algorithm possibly be? In this talk we will discuss the history of lower bounds for the effciency of online bin packing algorithms.

We start with a simple example showing that for any online bin packing algorithm $A$ we have $R_A^\infty\ge 4/3$. Then using the so-called Salzer-sequences we give combinatorial proofs to verify that $R_A^\infty\ge 1.5364$. Introducing an LP formulation van Vliet (1994) improved the lower bound to 1.5401. It was
an open question for a long time whether van Vliet's result can be proved in a purely
combinatorial way. In 2012 we gave a positive answer to this question.
Moreover, we managed to improve the lower bound to 1.5403 by using a different proof technique which is interesting in its own right, and also can be extended to arrive at an improved improved lower bound for a class of semi-online bin packing algorithms.

Abstract: We consider a simple symmetric random walk on the integers moving in random potential. The potentials represent a random risk of dying for the RW at each sites. A measure for the riskiness of moving in this random environment are the Lyapunov exponents. They observe the long time behaviour of the probability of reaching a certain site after starting at a fixed site. There are two different ways of treating the random potentials in this observation: the annealed (or averaged) and the quenched approach. We will see that in the case of the integers we can directly relate these two approaches to each other.

Random hyperbolic graphs have been suggested as a promising model of social networks. We consider the random hyperbolic graph model as formalized by Gugelmann et al. and essentially determine the spectral gap of the normalized Laplacian. Specifically, we establish that a.a.s.~the second smallest eigenvalue of the normalized Laplacian of the giant component of the random hyperbolic graph is $\Omega(n^{-(2\alpha-1)}/\polylog(n)$, where $\frac12<\alpha<1$ is a model parameter. As a byproduct we conclude that the conductance upper bound on the eigenvalue gap obtained via Cheeger's inequality is essentially tight. We also provide a more detailed picture of the collection of node on which the bound on the conductance is attained.

Joint work with Marcos Kiwi.

Strukturtheorie-Seminar

In this talk I will discuss one instance of the so called Hastings-Levitov planar aggregation model, consisting of growing random clusters on the complex plane, built by iterated composition of random conformal maps. In 2012 Norris and Turner proved that in the case of i.i.d. maps the limiting shape of these clusters is a disc. I will show that the fluctuations around this asymptotic behaviour are given by a random holomorphic Gaussian field F on $\{|z|>1\}$,
of which I will provide an explicit construction. The boundary values of F perform a Gaussian Markov process on the space of distributions, which is conveniently described as the solution of a stochastic PDE. When the cluster is allowed to grow indefinitely, this process converges to the restriction of the whole plane Gaussian Free Field to the unit circle.

Strukturtheorie-Seminar

We introduce a family of non Markovian nearest neighbor walks on $\mathbb{Z}$
called p-rotor walks, which depend on a parameter $p\in[0,1]$. For $p=1/2$ this
model is equivalent to simple random walk, and for $p=0$ we recover deterministic
rotor-router walk. We prove a functional limit law for p-rotor walk starting from
random i.i.d. rotor environment. The limit process is shown to be a doubly perturbed
Brownian motion $\mathcal{X}(t)$, which is the unique solution of the implicit
equation

We present asymptotic formulas and estimates for the Green function of isotropic unimodal convolution semigroups of probability measures on d-dimensional Euclidean space under the assumption that its Levy exponent varies regularly (including the case when it varies slowly).

The talk is based on a joint project with W. Cygan, M. Ryznar and B. Trojan.

Zahlentheoretisches Kolloquium

Abstract.} Let $k\in\N$, $p\in\PP$ and $a\in\Z$ such that $p\nmid a$. We consider a generalization of Legendre's symbol, the so called rational power residue symbol
\[
\left( \frac{a}{p} \right)_{2^k}:=\left\{\begin{array}{cl} 1, & \mbox{if there is $x\in\Z$ such that }x^{2^k}\equiv a \mod p, -1 & \mbox{else.} \end{array}\right.
\]
Let $N=pq$ be a semiprime number with large prime factors $p$ and $q$, $p\neq q$. The security of the RSA cryptosystem relies on the difficulty to compute $p$ and $q$ in the case that only $N$ is known. For $\gcd(N,a)=1$, we define $\left( \frac{a}{N} \right)_{2^k}:=\left( \frac{a}{p} \right)_{2^k}\cdot \left( \frac{a}{q} \right)_{2^k}$. In this talk, we will show that an efficient algorithm for computing this generalized Jacobi symbol would allow the efficient computation of the $2$-adic valuations $\nu_2(p-1)$ and $\nu_2(q-1)$ and, hence, of the first few bits of $p$ and $q$. We will also discuss the problem raised by this result, namely finding a reciprocity law for $\left( \frac{a}{N} \right)_{2^k}$.

Abstract}:
Let $q\ge 2$ be an integer and $\mathcal{E}$ be the set of prime numbers $p$ whose the sum of digits in base $q$ is equal to $\left\lfloor \frac{q-1}2 \frac{\log p}{\log q}\right\rfloor$. We prove that for every irrational number $\beta$, the sequence
$( \beta p)_{p\in\mathcal{E}}$ is uniformly distributed modulo 1.

This is a joint work with Christian Mauduit and Jo\"el Rivat (Universit\'e d'Aix-Marseille).

In the first part I shall describe briefly the classical approach to the study
of first passage times of random walks, which is based on the well-known duality lemma. This duality holds only for random walks with independent, identically distributed increments.

In the second part I shall present a new method, which allows one to study first passage times for a wider class of random walks. It turns out that, assuming that a random walk belongs to the domain of attraction of the Brownian motion, one can derive asymptotics for first passage times of random walk.

Abstract: Let $D$ be an integral domain of quotient field $K$, $A$
a torsion-free $D$-algebra and $B=A\otimes_D K$ the extended
$K$-algebra. The set of integer-valued polynomials on $A$ is
${\rm Int}(A)=\{f\in B[X] \mid f(A)\subseteq A\}$
and the intersection of ${\rm Int}(A)$ with $K[X]$ is the ring
${\rm Int}_K(A)$. In general, ${\rm Int}(A)$ is just a left
$\Int_K(A)$-module.

We present a definition of
${\rm Int}_K$-decomposable algebras (introduced recently by Nicholas Werner)
that is applicable also to $D$-algebras that are not free as
$D$-modules: we define $A$ to be ${\rm Int}_K$-decomposable when
${\rm Int}(A) \cong {\rm Int}_K(A) \otimes_D A$.

If $D$ is a Dedekind domain
with finite residue fields, we provide multiple characterizations of
such algebras: $A$ is ${\rm Int}_K$-decomposable algebras if and only if
either one of the following equivalent conditions hold:
\begin{enumerate}
\item
for each maximal ideal $P$ of $D$, $A/PA$ is a direct sum of copies
of a matrix ring over a finite field;
\item
for each maximal ideal $P$ of $D$, the
$P$-adic completion of $A$ is isomorphic to a finite direct sum of
copies of a matrix ring over a finite unramified extension of the
$P$-adic completion of $D$.
\end{enumerate}

We also prove that if $A$ is ${\rm Int}_K$-decomposable, then $A$
must be a maximal $D$-order in $B$ and $B$ must be separable
(hence, semisimple). Moreover, if $K$ is a number field with ring
of integers $D$, then the simple components of $B$ have the same
center, which is a finite unramified Galois extension of $K$.
This is joint work with Nicholas J. Werner.

We compute the precise (oscillating) asymptotic behavior of the diagonal elements of the heat kernel for the Vladimirov Laplacian on p-adic spaces. It turns out that it is a function of finite order which is not regularly varying at infinity.

We study Bonsall's cone spectral radius and the approximate point spectrum of (in general non-linear) positively homogeneous, bounded and supremum preserving maps, defined on a max-cone in a given normed vector lattice. We prove that Bonsall's cone spectral radius of such maps is always included in its approximate point spectrum. Our results apply to a large class of (nonlinear) max-type operators appearing in problems arising in certain differential and difference equations, mathematical physics and elsewhere, in particular to max type operators considered by Mallet-Parret and Nussbaum (2003).

By applying the techniques of proofs of the above result, we are also able to prove a generalization (to normed spaces and convex cones) of an (implicitly) known result that the spectral radius of a positive (linear) operator on a Banach lattice is contained in the approximate point spectrum. Under additional compactness type assumptions or for special cases of max-type operators we obtain additional results.

This is a joint work with Vladimir Müller (Institute of Mathematics, Czech Academy of Sciences).

Strukturtheorie-Seminar

In 1902 Burnside asked a question that was to become very influential in group theory: Can a group with all elements of finite order be infinite?

This question has found a positive answer in 1964 with the work of Golod and Shafarevich, but also in 1980 with a group generated by an automaton with only 5 states and 2 letters. Then, since the 80's, automaton groups arise as a powerful source of example of interesting groups.

In this talk we will discuss the impact of the structural properties of the underlying automaton on the generated group and present a combinatorial tool, the orbit tree of an automaton, that can be used to prove that some classes of automata cannot generate infinite Burnside groups. We will also give some idea of the proofs. No prior background is required.

In my talk I will introduce a one-dimensional percolation model dating back to Axelsson-Fisk and Häggström, and consider biased random walk on the infinite cluster of this model. Originally, this model has been designed to allow qualitative predictions of the behavior of biased random walk on the infinite cluster of supercritical percolation on $Z^d$

Häggström and Axelsson-Fisk have shown that biased random walk $(X_n)_n$ on the one-dimensional percolation cluster exhibits a phase transition: the linear speed $\lim_n X_n/n$ of the walk is positive iff the bias of the walk is strictly smaller than a critical bias $\lambda_c$.

I will explain that in the regime of the central limit theorem, that is, where the bias $\lambda$ is smaller than $\lambda_c/2$, the linear speed of the walk is a differentiable function of the bias $\lambda$.

{\bf Abstract:}
\[\]
Let ${(U_n)}_{n\geq 0}$ be a sequence of numbers given by $U_n=(\alpha^n - \beta^n)/(\alpha - \beta)$ for $n\geq 0$, where $\alpha$ and $\beta$
are complex numbers satisfying $|\beta|<1$ and $\alpha \beta =-1$. Let $s\geq 1$ and
\[\Phi_{2s} \,:=\, {(\alpha - \beta)}^{-2s} \sum_{n=1}^{\infty} \frac{1}{U_n^{2s}} \,.\]
In a joint paper of the lecturer with Prof. Shun Shimomura and Prof. Iekata Shiokawa it is shown that for any positive integers $s_1,s_2,s_3$
corresponding to algebraic $\alpha,\beta$ with $|\beta|<1$ and $\alpha \beta =-1$ the numbers $\Phi_{2s_1},\Phi_{2s_2},\Phi_{2s_3}$ are
algebraically independent over ${\Q}$ if and only if $s_1s_2s_3\equiv 0 \pmod 2$. In particular, for $\beta=(1-\sqrt{5})/2$ and $\beta=1-\sqrt{2}$
the results on $\Phi_{2s}$ are applicable to reciprocal sums formed by powers of Fibonacci numbers $U_n=F_n$ and by powers of Pell numbers $U_n=P_n$,
respectively. In the lecture the proof of the algebraic independence of $\Phi_{2s_1},\Phi_{2s_2},\Phi_{2s_3}$ is presented for the particular case
of three even integers $2\leq s_1<s_2<s_3$.

Übersichtsvortrag

Title: The mathematics of committee decisions: The impossibility of judgment aggregation and how to avoid itSpeaker: Prof. Christian List FBA (London School of Economics)Date: 7.3.2016, 16.15-17.45Room: STEG050, Steyerergasse 30, Erdgeschoss

Abstract:

How can the judgments of several individuals (committee members, judges,
voters) be aggregated into collective judgments, to be endorsed by the group as a whole? To illustrate that this is a non-trivial problem, suppose that a
multi-member court has to make a decision on some conclusion (``is the
defendant liable?'') on the basis of several premises (``was there a valid contract?'', ``was there a breach?''), where the premises are jointly necessary and sufficient for the conclusion (i.e., ``liability if and only if contract and breach''). It can then happen that each premise is supported by a
majority of the judges, while the conclusion is not. This shows that the court's judgment may depend on the voting protocol used. More generally, majority voting does not guarantee logically consistent collective judgments, even when all individual judgments are consistent.

The theory of judgment aggregation offers a framework for analysing such
collective decision problems. The aim of this talk is to give an
introductory overview of that theory and to explain some of its central
impossibility and possibility results.

No prior familiarity with this topic will be presupposed. For some
optional background reading, see:

\begin{abstract}
\newcommand{\bO}{\mathbf 0}
\newcommand{\bL}{\mathbf 1}
It is known that the number $a(\alpha,n)$ of binomial coefficients $\binom nt$ exactly divisible by $2^\alpha$ can be expressed using a polynomial $P_\alpha$ in block-additive functions in base $2$.
For example, we have
\[\begin{array}{ll}
a(1,n)/a(0,n)&=\tfrac 12\lvert n\rvert_{\bL\bO},
a(2,n)/a(0,n)&=-\frac 18\lvert n\rvert_{\bL\bO}+\lvert n\rvert_{\bL\bO\bO}+\frac 14\lvert n\rvert_{\bL\bL\bO}+\frac 18\lvert n\rvert_{\bL\bO}^2,
\end{array}\]
where $\lvert n\rvert_w$ is the number of times the word $w$ occurs as a subword of the base-$2$ expansion of $n$.

In this talk, we present a method for obtaining the sequence of coefficients of a given monomial in the polynomials $P_0,P_1,\ldots$ as a generating function.

In particular, we apply this method to the monomial $X_{\bL\bO}^k$ (corresponding to the term $\lvert n\rvert_{\bL\bO}^k$) and the associated sequence $\bigl(c^{(k)}_\alpha\bigr)_{\alpha\geq 0}$ of coefficients, obtaining the generating function
\[
\sum_{\alpha\geq 0}c^{(k)}_\alpha x^{\alpha}=\left(\log\left(1+\frac x2\right)\right)^k.
\]
\end{abstract}

In copositive optimization, it is essential to determine the minimal number of nonnegative vectors whose dyadic products form, summed up, a given completely positive matrix (indeed, one of these vectors necessarily must be a solution to the original problem). This matrix parameter is called cp-rank. Since long, it has been an open problem to determine the maximal possible cp-rank for any fixed order. Now we can refute a twenty years old conjecture and show that the known upper bounds are asymptotically equal to the lower ones.

Using techniques from symbolic dynamics, we prove a Garden of Eden type theorem for some hyperbolic homeomorphisms of a compact metrizable space (e.g. Anosov diffeomorphisms). This is a recent joint work with Michel Coornaert

Vortrag im Seminar für Kombinatorik und Optimierung

Recently Bringmann, Keusch, and Lengler introduced a new random graph model -- geometric inhomogeneous random graphs (GIRGs) -- possessing many of the characteristic properties of complex real-world networks (Power-law degree distribution, small diameter, etc.). Each vertex consists of a geometric position and a weight and any two vertices $u,v$ are connected by an edge independently with some probability $p_{u,v}$ depending on their weights and relative position. In particular, GIRGs generalise the well-studied model of hyperbolic random graphs and facilitate their analysis by stripping off most of the technical details.

In this talk we investigate the evolution of a well-known infection process called bootstrap percolation with localised initial infection on GIRGs. Given some infection parameter $r>1$ and a ball $B_0$ of volume $\nu$ we initially infect any vertex lying in $B_0$ with some probability $p_0$. And then round after round, the infection spreads to all those still uninfected vertices which have at least $r$ infected neighbors. Once a vertex becomes infected, it remains so forever. We show that this process exhibits a threshold behaviour in terms of $p_0$, in other words, there is a critical probability $p_c=p_c(\nu)$ such that the following holds: If $p_0/p_c\to\infty$ then the process starts to accelerate very quickly and eventually infects a large proportion of all vertices; while if $p_0/p_c\to 0$ then it stops without infecting substantially more vertices than those which were infected initially.

This extends some results by Candellero and Fountoulakis on bootstrap percolation with global initial infection on hyperbolic random graphs.

A classical result states that the number of closed walks of length k in a graph G is equal to the trace of the k-th power of its adjacency matrix A. If G is connected, its largest eigenvalue $\lambda_1$ has maximum absolute value among all eigenvalues of A by the Perron-Frobenius theorem, so that
$$
\lambda_1=\lim_{k\to\infty}\sqrt[2k]{\mathrm{tr}(A^{2k})}.
$$
Let $G\preccurlyeq H$ denote that the number of closed walks of length~$k$ in~$G$ is less than or equal to the number of closed walks of length~$k$ in~$H$ for infinitely many values of~$k$. Hence $G\preccurlyeq H\,\Rightarrow\,\lambda_{G,1}\leq\lambda_{H,1}$.

We present here a general graph composition, the so-called {\em multiple coalescence}, which enables one to count closed walks in composed graph in terms of closed walk counts of constituent graphs. As a result, multiple coalescence preserves the relation $\preccurlyeq$ under natural and mild assumptions on its constituents. We will showcase the use of multiple coalescence in the resolution of Belardo, Li Marzi and Simi\'c's conjectured generalizations of the Li-Feng lemmas and apply them to study spectral radius of rooted product of graphs. We will also discuss current obstacles in applying multiple coalescence to attack the 1991 conjecture of Cvetkovi\'c and Rowlinson on the maximum spectral radius of outerplanar graphs.